which is an injective trivial cofibration.
Since \({H}_\infty ^i\)  is a symmetric monoidal model category, and since \(\Delta _{e!}\)  preserves pushouts, it now follows that \(\Delta _{e!}\)  sends the generating (trivial) cofibrations of \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  to (trivial) cofibrations in \({H}_\infty ^i\) .
Thus, \(\Delta _{e!}\)  is a left Quillen functor by [1]}.
